A Comprehensive Evaluation of the DFP Method for Geometric Constraint Solving Algorithm Using PlaneGCS

The development of open-source geometric constraint solvers is a pressing research topic, as commercially available solvers may not meet the research requirements. In this paper, we examine the use of numerical methods in PlaneGCS, an open-source geometric constraint solver within the FreeCAD CAD software. Our study focuses on PlaneGCS\u27s constraint solving algorithms and the three built-in single-subsystem solving methods: BFGS, LM, and Dogleg. Based on our research results, the DFP method was implemented in PlaneGCS and was successfully verified in FreeCAD. To evaluate the performance of the algorithms, we used the solving state of the constraint system as a test criterion, and analysed their solving time, adaptability, and number of iterations. Our results highlight the performance differences between the algorithms and provide empirical guidance for selection of constraint solving algorithms and research based on open-source geometric constraint solvers

An interactive N-Dimensional constraint system

Journal ArticleIn this paper, we present a graph-based approach to geometric constraint solving. Geometric primitives (points, lines, circles, planes, etc.) possess intrinsic degrees of freedom in their embedding space. Constraints reduce the degrees of freedom of a set of objects. A constraint graph is created with objects as the nodes, and the constraints as the arcs. A graph algorithm transforms the undirected constraint graph into a directed acyclic dependency graph which can be directly used to derive a sequence of construction operations as a symbolic solution to the constraint problem. The approach has been generalized to an n-dimensional space, which, among other things, allows for a uniform handling of 2-D and 3-D constraint problems or algebraic constraints between scalar dimension. Solutions of arbitrary dimensions can be interpreted as approaches to over- and under- constrained problems. In this paper, we present the theoretical background of the approach, and report the results of it's application within an interactive modeling system

Moving into higher dimensions of geometric constraint solving

Journal ArticleIn this paper, we present an approach to geometric constraint solving, based on degree of freedom analysis. Any geometric primitive (point, line, circle, plane, etc.) possesses an intrinsic degree of freedom in its embedding space which is usually two or three dimensional. Constraints reduce the degrees of freedom of an object (or a set of objects). We use graph algorithms to determine upper and lower bounds for the degrees of freedom of a set of constrained objects, symbolically. This analysis is then used to establish dependency graphs and evaluation schemes for symbolic or numeric solutions to constraint problems. The approach has been generalized for n-dimensional space, which, among other things, allows for a uniform handling of 2-D and 3-D constraint problems or algebraic constraints between scalar dimension. Also, higher than three dimensional solutions can be interpreted as approaches to over- and under- constrained problems. In this paper, we will present the theoretical background of the approach, and demonstrate how it can be applied within an interactive design environment

Review of dimensioning and tolerancing: representation and processing

The paper surveys the current state of knowledge of techniques for representing, manipulating and analysing dimensioning and tolerancing data in computer-aided design and manufacturing. The use of solid models and variational geometry, and its implications for the successful integration of CAD and CAM, are discussed. The topics explored so far can be grouped into four categories: (a) the representation of dimensioning and tolerancing (D & T), (b) the synthesis and analysis of D & T, (c) tolerance control, and (d) the implications of D & T in CAM. The paper describes in detail the recent work in each group, and concludes with speculation on a general framework for future research.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29159/1/0000204.pd

Symbolic dimensioning in computer-aided design

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1980.MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING.Bibliography: leaves 89-90.by Robert Allan Light.M.S

Direct tree decomposition of geometric constraint graphs

The evolution of constraint based geometric models is tightly tied to parametric and feature-based Computer-Aided Design (CAD) systems. Since the introduction of parametric design by Pro/Engineer in the 1980's, most major CAD systems adopted constraint based geometric models as a core technology. Constraint based geometric models allowed CAD systems to provide a more powerful data model while offering an intuitive user interface. Later on, the same models also found application to fields like linkage design, chemical modeling, computer vision and dynamic geometry. Constraint based geometric models are unevaluated models. A key problem related to constraint based geometric models is the geometric constraint based solving problem which, roughly speaking, can be stated as the problem of evaluating a constraint based model. Among the different approaches to geometric constraint solving, we are interested in graph-based Decomposition-Recombination solvers. In the graph-based constructive approach, the geometric problem is first translated into a graph whose vertices represent the set of geometric elements and whose edges are the constraints. Then the constraint problem is solved by decomposing the graph into a set of sub-problems, each sub-problem is recursively divided until reaching basic problems which are solved by a dedicated equational solver. The solution to the initial problem is computed by merging the solutions to the sub-problems. The approach used by DR-solvers has been particularly successful when the decomposition into subproblems and subsequent recombination of solutions to these subproblems can be described by a plan generated a priori, that is, a plan generated as a preprocessing step without actually solving the subsystems. The plan output by the DR-planner remains unchanged as numerical values of parameters change. Such a plan is known as a DR-plan and the unit in the solver that generates it is the DR-planner. In this setting, the DR-plan is then used to drive the actual solving process, that is, computing specific coordinates that properly place geometric objects with respect to each other. In this thesis we develop a new DR-planner algorithm for graph-constructive two dimensional DR-solvers. This DR-planner is based on the tree-decomposition of a graph. The triangle- or tree-decomposition of a graph decomposes a graph into three subgraphs such that subgraphs pairwise share one vertex. Shared vertices are called hinges. The tree-decomposition of a geometric constraint graph is in some sense the construction plan that solves the corresponding problem. The DR-planner algorithm first transforms the input graph into a simpler, planar graph. After that, an specific planar embedding is computed for the transformed graph where hinges, if any, can be straightly found. In the work we proof the soundness of the new algorithm. We also show that the worst case time performance of the resthe number of vertices of the input graph. The resulting algorithm is easy to implement and is as efficient as other known solving algorithms.L'evolució de models geomètrics basats en restriccions està fortament lligada al sistemes de Disseny Assistit per Computador (CAD) paramètrics i als basats en el paradigma de disseny per mitjà de característiques. Des de la introducció del disseny paramètric per part de Pro/Engineer en els anys 80, la major part de sistemes CAD utilitzaren com a tecnologia de base els models geomètrics basats en restriccions. Els models geomètrics basats en restriccions permeteren als sistemes CAD proporcionar un model d'informació més ampli i alhora oferir una interfície d'usuari intuitiva. Posteriorment, els mateixos models s'aplicaren en camps com el disseny de mecanismes, el modelatge químic, la visió per computador i la geometria dinàmica. Els models geomètrics basats en restriccions són models no avaluats. Un problema clau relacionat amb el models de restriccions geomètriques és el problema de la resolució de restriccions geomètriques, que es resumeix com el problema d'avaluar un model basat en restriccions. Entre els diferents enfocs de resolució de restriccions geomètriques, tractem els solvers de Descomposició-Recombinació (DR-solvers) basats en graphs. En l'enfoc constructiu basat en grafs, el problema geomètric es trasllada en un pas inicial a un graf, on els vèrtexs del graf representen el conjunt d'elements geomètrics i on les arestes corresponen a les restriccions geomètriques entre els elements. A continuació el problema de restriccions es resol descomposant el graf en un conjunt de subproblemes, cadascun dels quals es divideix recursivament fins a obtenir problemes bàsics, que sovint són operacions geomètriques realitzables, per exemple, amb regle i compàs, i que es resolen per mitjà d'un solver numèric específic. Finalment, la solució del problema inicial s'obté recombinant les solucions dels subproblemes. L'enfoc utilitzat pels DR-solvers ha esdevingut especialment interessant quan la descomposició en subproblemes i la posterior recombinació de solucions d'aquests subproblemes es pot descriure com un pla de construcció generat a priori, és a dir, un pla generat com a pas de pre-procés sense necessitat de resoldre realment els subsistemes. El pla generat pel DR-planner esdevé inalterable encara que els valors numèrics dels paràmetres canviin. Aquest pla es coneix com a DR-plan i la unitat en el solver que el genera és l'anomenat DR-planner. En aquest context, el DR-plan s'utilitza com a eina del procés de resolució en curs, és a dir, permet calcular les coordenades específiques que correctament posicionen els elements geomètrics uns respecte els altres. En aquesta tesi desenvolupem un nou algoritme que és la base del DR-planner per a DR-solvers constructius basats en grafs en l'espai bidimensional. Aquest DR-planner es basa en la descomposició en arbre d'un graf. La descomposició en triangles o arbre de descomposició d'un graf es basa en descomposar un graf en tres subgrafs tals que comparteixen un vèrtex 2 a 2. El conjunt de vèrtexs compartits s'anomenen \emph{hinges}. La descomposició en arbre d'un graf de restriccions geomètriques equival, en cert sentit, a resoldre el problema de restriccions geomètriques. L'algoritme del DR-planner en primer lloc transforma el graf proporcionat en un graf més simple i planar. A continuació, es calcula el dibuix en el pla del graf transformat, on les hinges, si n'hi ha, es calculen de manera directa. En aquest treball demostrem la correctesa del nou algoritme. Finalment, proporcionem l'estudi de la complexitat temporal de l'algoritme en cas pitjor i demostrem que és quadràtica en el nombre de vèrtexs del graf proporcionat. L'algoritme resultant és senzill d'implementar i tan eficient com altres algoritmes de resolució concret

Methods for computer aided inspection of geometric tolerances

This thesis investigates computational methods for assessing tolerance specifications of geometric features in a context of computer aided inspection. It is concerned with checking the sampled features for containment within tolerance zones specified at the design stage, not with explicit shape measurement. The significance of this difference is highlighted when two or more features are to be inspected in combination. The approach adopted is to express the tolerance information as a set of inequality constraints and then to seek efficient methods for determining the feasibility of the set, that is whether all the constraints can be simultaneously satisfied. Roundness inspection is used to introduce all the concepts of the new formulations. By linearisation of the constraints, a standard approximation in roundness measurement, a new algorithm is implemented which provides a “GO-NOGO” result of inspection by checking for feasibility in a highly efficient way. This algorithmic approach is then extended to other inspection situations where naturally linear constraints or valid linearisation occur. Since there are many inspection cases where linearisation is not appropriate, non-linear optimisation techniques are then investigated for their effectiveness in feasibility testing. The inspection of arrays of circular features is used here as a typical test case. Genetic search methods are explored as a possible alternative to formal non-linear programming and guidelines for their efficient use for this problem are proposed. These methods are then compared and contrasted with formal methods, particularly generalised reduced gradient (GRG) and sequential quadratic programming (SQP). The linear algorithm is shown to be the most efficient when it can be used, although all techniques were fast enough for on-line use with modest sized data sets. Currently all the non-linear methods are too expensive for routine use on large data sets. GRG is recommended as having the most favourable combination of good and bad features, but there is some evidence that genetic search might be relatively more efficient for more complex inspection problems

Modelling and controlling variation propagation in mechanical assembly of high speed rotating machines

Assembly plays a vital role in the quality of a final product and has a great impact on the manufacturing cost. The mechanical assemblies consist of parts that inevitably have variations from their ideal dimensions. These variations propagate and accumulate as parts are assembled together. Excessive amount of variations in an assembly may cause improper functionality of the product being assembled. Improving assembly quality and reducing the assembly time and cost are the main objectives of this thesis. The quality of an assembly is determined in terms of variations in critical assembly dimensions, also known as Key Characteristics (KCs). Key Characteristics are designated to indicate where excess variation will affect product quality and what product features and tolerances require special attention. In order to improve assembly quality and reduce assembly time and cost, it is necessary to: (1) model non-ideal parts based on tolerances defined in design standards or current industrial practice of component inspection, (2) model assemblies and their associated assembly processes to analyse tolerance stack-up in the assembly, (3) develop probabilistic model to predict assembly variation after product assembly, and (4) implement control strategies for minimising assembly variation propagations to find optimum configuration of the assembly. Two assembly models have been developed, a linear model and a fully non-linear model for calculating assembly variation propagations. The assembly models presented in this thesis also allows for inclusion of geometric feature variation of each assembly component. Methods of incorporating geometric feature variations into an assembly variation model are described and analysis techniques are explained. The assembly variation model and the geometric variation models have been developed for 20 and 3D assemblies. Modelling techniques for incorporating process and measurement noise are also developed and described for the nonlinear assembly model and results are given to demonstrate the calculation of assembly variations while considering part, process and measurement errors. Two assembly case studies originating in sub-assemblies of aero-engines have been studied: Case Study 1, representing the rotating part (rotor) of an aero-engine, and Case Study 2, representing non-rotating part (stator) of an aero-engine. A probabilistic method based on the linear model is presented as a general analytical method for analysis of 3D mechanical assemblies. Probability density functions are derived for assembly position errors to analyse a general mechanical assembly, and separate probability functions are derived for the Key Characteristics (KCs) for assembly in Case Studies 1 and 2. The derived probability functions are validated by using the Monte Carlo simulation method based on the exact (full non-linear) model. Results showed that the proposed probabilistic method of estimating tolerance accumulation in mechanical assemblies is very efficient and accurate when compared to the Monte Carlo simulation method, particularly if large variations at the tails of the distributions are considered. Separate control strategies have been implemented for each case study. Four methods are proposed to minimise assembly variations for Case Study 1, and one error minimisation method is suggested for assemblies of Case Study 2. Based on the developed methods to optimise assembly quality, the two case studies were investigated, and it was found that the proposed optimisation methods can significantly improve assembly quality. The developed optimisation methods do not require any special tooling (such as fixtures) and can easily be implemented in practice